Source code for statsmodels.graphics.functional

"""Module for functional boxplots."""
import numpy as np
from scipy.special import comb

from statsmodels.graphics.utils import _import_mpl
from statsmodels.multivariate.pca import PCA
from statsmodels.nonparametric.kernel_density import KDEMultivariate

try:
    from scipy.optimize import brute, differential_evolution, fmin
    have_de_optim = True
except ImportError:
    from scipy.optimize import brute, fmin
    have_de_optim = False
import itertools
from multiprocessing import Pool

from . import utils

__all__ = ['hdrboxplot', 'fboxplot', 'rainbowplot', 'banddepth']


class HdrResults(object):
    """Wrap results and pretty print them."""

    def __init__(self, kwds):
        self.__dict__.update(kwds)

    def __repr__(self):
        msg = ("HDR boxplot summary:\n"
               "-> median:\n{}\n"
               "-> 50% HDR (max, min):\n{}\n"
               "-> 90% HDR (max, min):\n{}\n"
               "-> Extra quantiles (max, min):\n{}\n"
               "-> Outliers:\n{}\n"
               "-> Outliers indices:\n{}\n"
               ).format(self.median, self.hdr_50, self.hdr_90,
                        self.extra_quantiles, self.outliers, self.outliers_idx)

        return msg


def _inverse_transform(pca, data):
    """
    Inverse transform on PCA.

    Use PCA's `project` method by temporary replacing its factors with
    `data`.

    Parameters
    ----------
    pca : statsmodels Principal Component Analysis instance
        The PCA object to use.
    data : sequence of ndarrays or 2-D ndarray
        The vectors of functions to create a functional boxplot from.  If a
        sequence of 1-D arrays, these should all be the same size.
        The first axis is the function index, the second axis the one along
        which the function is defined.  So ``data[0, :]`` is the first
        functional curve.

    Returns
    -------
    projection : ndarray
        nobs by nvar array of the projection onto ncomp factors
    """
    factors = pca.factors
    pca.factors = data.reshape(-1, factors.shape[1])
    projection = pca.project()
    pca.factors = factors
    return projection


def _curve_constrained(x, idx, sign, band, pca, ks_gaussian):
    """Find out if the curve is within the band.

    The curve value at :attr:`idx` for a given PDF is only returned if
    within bounds defined by the band. Otherwise, 1E6 is returned.

    Parameters
    ----------
    x : float
        Curve in reduced space.
    idx : int
        Index value of the components to compute.
    sign : int
        Return positive or negative value.
    band : list of float
        PDF values `[min_pdf, max_pdf]` to be within.
    pca : statsmodels Principal Component Analysis instance
        The PCA object to use.
    ks_gaussian : KDEMultivariate instance

    Returns
    -------
    value : float
        Curve value at `idx`.
    """
    x = x.reshape(1, -1)
    pdf = ks_gaussian.pdf(x)
    if band[0] < pdf < band[1]:
        value = sign * _inverse_transform(pca, x)[0][idx]
    else:
        value = 1E6
    return value


def _min_max_band(args):
    """
    Min and max values at `idx`.

    Global optimization to find the extrema per component.

    Parameters
    ----------
    args: list
        It is a list of an idx and other arguments as a tuple:
            idx : int
                Index value of the components to compute
        The tuple contains:
            band : list of float
                PDF values `[min_pdf, max_pdf]` to be within.
            pca : statsmodels Principal Component Analysis instance
                The PCA object to use.
            bounds : sequence
                ``(min, max)`` pair for each components
            ks_gaussian : KDEMultivariate instance

    Returns
    -------
    band : tuple of float
        ``(max, min)`` curve values at `idx`
    """
    idx, (band, pca, bounds, ks_gaussian, use_brute, seed) = args
    if have_de_optim and not use_brute:
        max_ = differential_evolution(_curve_constrained, bounds=bounds,
                                      args=(idx, -1, band, pca, ks_gaussian),
                                      maxiter=7, seed=seed).x
        min_ = differential_evolution(_curve_constrained, bounds=bounds,
                                      args=(idx, 1, band, pca, ks_gaussian),
                                      maxiter=7, seed=seed).x
    else:
        max_ = brute(_curve_constrained, ranges=bounds, finish=fmin,
                     args=(idx, -1, band, pca, ks_gaussian))

        min_ = brute(_curve_constrained, ranges=bounds, finish=fmin,
                     args=(idx, 1, band, pca, ks_gaussian))

    band = (_inverse_transform(pca, max_)[0][idx],
            _inverse_transform(pca, min_)[0][idx])
    return band


[docs]def hdrboxplot(data, ncomp=2, alpha=None, threshold=0.95, bw=None, xdata=None, labels=None, ax=None, use_brute=False, seed=None): """ High Density Region boxplot Parameters ---------- data : sequence of ndarrays or 2-D ndarray The vectors of functions to create a functional boxplot from. If a sequence of 1-D arrays, these should all be the same size. The first axis is the function index, the second axis the one along which the function is defined. So ``data[0, :]`` is the first functional curve. ncomp : int, optional Number of components to use. If None, returns the as many as the smaller of the number of rows or columns in data. alpha : list of floats between 0 and 1, optional Extra quantile values to compute. Default is None threshold : float between 0 and 1, optional Percentile threshold value for outliers detection. High value means a lower sensitivity to outliers. Default is `0.95`. bw : array_like or str, optional If an array, it is a fixed user-specified bandwidth. If `None`, set to `normal_reference`. If a string, should be one of: - normal_reference: normal reference rule of thumb (default) - cv_ml: cross validation maximum likelihood - cv_ls: cross validation least squares xdata : ndarray, optional The independent variable for the data. If not given, it is assumed to be an array of integers 0..N-1 with N the length of the vectors in `data`. labels : sequence of scalar or str, optional The labels or identifiers of the curves in `data`. If not given, outliers are labeled in the plot with array indices. ax : AxesSubplot, optional If given, this subplot is used to plot in instead of a new figure being created. use_brute : bool Use the brute force optimizer instead of the default differential evolution to find the curves. Default is False. seed : {None, int, np.random.RandomState} Seed value to pass to scipy.optimize.differential_evolution. Can be an integer or RandomState instance. If None, then the default RandomState provided by np.random is used. Returns ------- fig : Figure If `ax` is None, the created figure. Otherwise the figure to which `ax` is connected. hdr_res : HdrResults instance An `HdrResults` instance with the following attributes: - 'median', array. Median curve. - 'hdr_50', array. 50% quantile band. [sup, inf] curves - 'hdr_90', list of array. 90% quantile band. [sup, inf] curves. - 'extra_quantiles', list of array. Extra quantile band. [sup, inf] curves. - 'outliers', ndarray. Outlier curves. See Also -------- banddepth, rainbowplot, fboxplot Notes ----- The median curve is the curve with the highest probability on the reduced space of a Principal Component Analysis (PCA). Outliers are defined as curves that fall outside the band corresponding to the quantile given by `threshold`. The non-outlying region is defined as the band made up of all the non-outlying curves. Behind the scene, the dataset is represented as a matrix. Each line corresponding to a 1D curve. This matrix is then decomposed using Principal Components Analysis (PCA). This allows to represent the data using a finite number of modes, or components. This compression process allows to turn the functional representation into a scalar representation of the matrix. In other words, you can visualize each curve from its components. Each curve is thus a point in this reduced space. With 2 components, this is called a bivariate plot (2D plot). In this plot, if some points are adjacent (similar components), it means that back in the original space, the curves are similar. Then, finding the median curve means finding the higher density region (HDR) in the reduced space. Moreover, the more you get away from this HDR, the more the curve is unlikely to be similar to the other curves. Using a kernel smoothing technique, the probability density function (PDF) of the multivariate space can be recovered. From this PDF, it is possible to compute the density probability linked to the cluster of points and plot its contours. Finally, using these contours, the different quantiles can be extracted along with the median curve and the outliers. Steps to produce the HDR boxplot include: 1. Compute a multivariate kernel density estimation 2. Compute contour lines for quantiles 90%, 50% and `alpha` % 3. Plot the bivariate plot 4. Compute median curve along with quantiles and outliers curves. References ---------- [1] R.J. Hyndman and H.L. Shang, "Rainbow Plots, Bagplots, and Boxplots for Functional Data", vol. 19, pp. 29-45, 2010. Examples -------- Load the El Nino dataset. Consists of 60 years worth of Pacific Ocean sea surface temperature data. >>> import matplotlib.pyplot as plt >>> import statsmodels.api as sm >>> data = sm.datasets.elnino.load() Create a functional boxplot. We see that the years 1982-83 and 1997-98 are outliers; these are the years where El Nino (a climate pattern characterized by warming up of the sea surface and higher air pressures) occurred with unusual intensity. >>> fig = plt.figure() >>> ax = fig.add_subplot(111) >>> res = sm.graphics.hdrboxplot(data.raw_data[:, 1:], ... labels=data.raw_data[:, 0].astype(int), ... ax=ax) >>> ax.set_xlabel("Month of the year") >>> ax.set_ylabel("Sea surface temperature (C)") >>> ax.set_xticks(np.arange(13, step=3) - 1) >>> ax.set_xticklabels(["", "Mar", "Jun", "Sep", "Dec"]) >>> ax.set_xlim([-0.2, 11.2]) >>> plt.show() .. plot:: plots/graphics_functional_hdrboxplot.py """ fig, ax = utils.create_mpl_ax(ax) if labels is None: # For use with pandas, get the labels if hasattr(data, 'index'): labels = data.index else: labels = np.arange(len(data)) data = np.asarray(data) if xdata is None: xdata = np.arange(data.shape[1]) n_samples, dim = data.shape # PCA and bivariate plot pca = PCA(data, ncomp=ncomp) data_r = pca.factors # Create gaussian kernel ks_gaussian = KDEMultivariate(data_r, bw=bw, var_type='c' * data_r.shape[1]) # Boundaries of the n-variate space bounds = np.array([data_r.min(axis=0), data_r.max(axis=0)]).T # Compute contour line of pvalue linked to a given probability level if alpha is None: alpha = [threshold, 0.9, 0.5] else: alpha.extend([threshold, 0.9, 0.5]) alpha = list(set(alpha)) alpha.sort(reverse=True) n_quantiles = len(alpha) pdf_r = ks_gaussian.pdf(data_r).flatten() pvalues = [np.percentile(pdf_r, (1 - alpha[i]) * 100, interpolation='linear') for i in range(n_quantiles)] # Find mean, outliers curves if have_de_optim and not use_brute: median = differential_evolution(lambda x: - ks_gaussian.pdf(x), bounds=bounds, maxiter=5, seed=seed).x else: median = brute(lambda x: - ks_gaussian.pdf(x), ranges=bounds, finish=fmin) outliers_idx = np.where(pdf_r < pvalues[alpha.index(threshold)])[0] labels_outlier = [labels[i] for i in outliers_idx] outliers = data[outliers_idx] # Find HDR given some quantiles def _band_quantiles(band, use_brute=use_brute, seed=seed): """ Find extreme curves for a quantile band. From the `band` of quantiles, the associated PDF extrema values are computed. If `min_alpha` is not provided (single quantile value), `max_pdf` is set to `1E6` in order not to constrain the problem on high values. An optimization is performed per component in order to find the min and max curves. This is done by comparing the PDF value of a given curve with the band PDF. Parameters ---------- band : array_like alpha values ``(max_alpha, min_alpha)`` ex: ``[0.9, 0.5]`` use_brute : bool Use the brute force optimizer instead of the default differential evolution to find the curves. Default is False. seed : {None, int, np.random.RandomState} Seed value to pass to scipy.optimize.differential_evolution. Can be an integer or RandomState instance. If None, then the default RandomState provided by np.random is used. Returns ------- band_quantiles : list of 1-D array ``(max_quantile, min_quantile)`` (2, n_features) """ min_pdf = pvalues[alpha.index(band[0])] try: max_pdf = pvalues[alpha.index(band[1])] except IndexError: max_pdf = 1E6 band = [min_pdf, max_pdf] pool = Pool() data = zip(range(dim), itertools.repeat((band, pca, bounds, ks_gaussian, seed, use_brute))) band_quantiles = pool.map(_min_max_band, data) pool.terminate() pool.close() band_quantiles = list(zip(*band_quantiles)) return band_quantiles extra_alpha = [i for i in alpha if 0.5 != i and 0.9 != i and threshold != i] if len(extra_alpha) > 0: extra_quantiles = [] for x in extra_alpha: for y in _band_quantiles([x], use_brute=use_brute, seed=seed): extra_quantiles.append(y) else: extra_quantiles = [] # Inverse transform from n-variate plot to dataset dataset's shape median = _inverse_transform(pca, median)[0] hdr_90 = _band_quantiles([0.9, 0.5], use_brute=use_brute, seed=seed) hdr_50 = _band_quantiles([0.5], use_brute=use_brute, seed=seed) hdr_res = HdrResults({ "median": median, "hdr_50": hdr_50, "hdr_90": hdr_90, "extra_quantiles": extra_quantiles, "outliers": outliers, "outliers_idx": outliers_idx }) # Plots ax.plot(np.array([xdata] * n_samples).T, data.T, c='c', alpha=.1, label=None) ax.plot(xdata, median, c='k', label='Median') fill_betweens = [] fill_betweens.append(ax.fill_between(xdata, *hdr_50, color='gray', alpha=.4, label='50% HDR')) fill_betweens.append(ax.fill_between(xdata, *hdr_90, color='gray', alpha=.3, label='90% HDR')) if len(extra_quantiles) != 0: ax.plot(np.array([xdata] * len(extra_quantiles)).T, np.array(extra_quantiles).T, c='y', ls='-.', alpha=.4, label='Extra quantiles') if len(outliers) != 0: for ii, outlier in enumerate(outliers): if labels_outlier is None: label = 'Outliers' else: label = str(labels_outlier[ii]) ax.plot(xdata, outlier, ls='--', alpha=0.7, label=label) handles, labels = ax.get_legend_handles_labels() # Proxy artist for fill_between legend entry # See https://matplotlib.org/1.3.1/users/legend_guide.html plt = _import_mpl() for label, fill_between in zip(['50% HDR', '90% HDR'], fill_betweens): p = plt.Rectangle((0, 0), 1, 1, fc=fill_between.get_facecolor()[0]) handles.append(p) labels.append(label) by_label = dict(zip(labels, handles)) if len(outliers) != 0: by_label.pop('Median') by_label.pop('50% HDR') by_label.pop('90% HDR') ax.legend(by_label.values(), by_label.keys(), loc='best') return fig, hdr_res
[docs]def fboxplot(data, xdata=None, labels=None, depth=None, method='MBD', wfactor=1.5, ax=None, plot_opts=None): """ Plot functional boxplot. A functional boxplot is the analog of a boxplot for functional data. Functional data is any type of data that varies over a continuum, i.e. curves, probability distributions, seasonal data, etc. The data is first ordered, the order statistic used here is `banddepth`. Plotted are then the median curve, the envelope of the 50% central region, the maximum non-outlying envelope and the outlier curves. Parameters ---------- data : sequence of ndarrays or 2-D ndarray The vectors of functions to create a functional boxplot from. If a sequence of 1-D arrays, these should all be the same size. The first axis is the function index, the second axis the one along which the function is defined. So ``data[0, :]`` is the first functional curve. xdata : ndarray, optional The independent variable for the data. If not given, it is assumed to be an array of integers 0..N-1 with N the length of the vectors in `data`. labels : sequence of scalar or str, optional The labels or identifiers of the curves in `data`. If given, outliers are labeled in the plot. depth : ndarray, optional A 1-D array of band depths for `data`, or equivalent order statistic. If not given, it will be calculated through `banddepth`. method : {'MBD', 'BD2'}, optional The method to use to calculate the band depth. Default is 'MBD'. wfactor : float, optional Factor by which the central 50% region is multiplied to find the outer region (analog of "whiskers" of a classical boxplot). ax : AxesSubplot, optional If given, this subplot is used to plot in instead of a new figure being created. plot_opts : dict, optional A dictionary with plotting options. Any of the following can be provided, if not present in `plot_opts` the defaults will be used:: - 'cmap_outliers', a Matplotlib LinearSegmentedColormap instance. - 'c_inner', valid MPL color. Color of the central 50% region - 'c_outer', valid MPL color. Color of the non-outlying region - 'c_median', valid MPL color. Color of the median. - 'lw_outliers', scalar. Linewidth for drawing outlier curves. - 'lw_median', scalar. Linewidth for drawing the median curve. - 'draw_nonout', bool. If True, also draw non-outlying curves. Returns ------- fig : Figure If `ax` is None, the created figure. Otherwise the figure to which `ax` is connected. depth : ndarray A 1-D array containing the calculated band depths of the curves. ix_depth : ndarray A 1-D array of indices needed to order curves (or `depth`) from most to least central curve. ix_outliers : ndarray A 1-D array of indices of outlying curves in `data`. See Also -------- banddepth, rainbowplot Notes ----- The median curve is the curve with the highest band depth. Outliers are defined as curves that fall outside the band created by multiplying the central region by `wfactor`. Note that the range over which they fall outside this band does not matter, a single data point outside the band is enough. If the data is noisy, smoothing may therefore be required. The non-outlying region is defined as the band made up of all the non-outlying curves. References ---------- [1] Y. Sun and M.G. Genton, "Functional Boxplots", Journal of Computational and Graphical Statistics, vol. 20, pp. 1-19, 2011. [2] R.J. Hyndman and H.L. Shang, "Rainbow Plots, Bagplots, and Boxplots for Functional Data", vol. 19, pp. 29-45, 2010. Examples -------- Load the El Nino dataset. Consists of 60 years worth of Pacific Ocean sea surface temperature data. >>> import matplotlib.pyplot as plt >>> import statsmodels.api as sm >>> data = sm.datasets.elnino.load() Create a functional boxplot. We see that the years 1982-83 and 1997-98 are outliers; these are the years where El Nino (a climate pattern characterized by warming up of the sea surface and higher air pressures) occurred with unusual intensity. >>> fig = plt.figure() >>> ax = fig.add_subplot(111) >>> res = sm.graphics.fboxplot(data.raw_data[:, 1:], wfactor=2.58, ... labels=data.raw_data[:, 0].astype(int), ... ax=ax) >>> ax.set_xlabel("Month of the year") >>> ax.set_ylabel("Sea surface temperature (C)") >>> ax.set_xticks(np.arange(13, step=3) - 1) >>> ax.set_xticklabels(["", "Mar", "Jun", "Sep", "Dec"]) >>> ax.set_xlim([-0.2, 11.2]) >>> plt.show() .. plot:: plots/graphics_functional_fboxplot.py """ fig, ax = utils.create_mpl_ax(ax) plot_opts = {} if plot_opts is None else plot_opts if plot_opts.get('cmap_outliers') is None: from matplotlib.cm import rainbow_r plot_opts['cmap_outliers'] = rainbow_r data = np.asarray(data) if xdata is None: xdata = np.arange(data.shape[1]) # Calculate band depth if required. if depth is None: if method not in ['MBD', 'BD2']: raise ValueError("Unknown value for parameter `method`.") depth = banddepth(data, method=method) else: if depth.size != data.shape[0]: raise ValueError("Provided `depth` array is not of correct size.") # Inner area is 25%-75% region of band-depth ordered curves. ix_depth = np.argsort(depth)[::-1] median_curve = data[ix_depth[0], :] ix_IQR = data.shape[0] // 2 lower = data[ix_depth[0:ix_IQR], :].min(axis=0) upper = data[ix_depth[0:ix_IQR], :].max(axis=0) # Determine region for outlier detection inner_median = np.median(data[ix_depth[0:ix_IQR], :], axis=0) lower_fence = inner_median - (inner_median - lower) * wfactor upper_fence = inner_median + (upper - inner_median) * wfactor # Find outliers. ix_outliers = [] ix_nonout = [] for ii in range(data.shape[0]): if (np.any(data[ii, :] > upper_fence) or np.any(data[ii, :] < lower_fence)): ix_outliers.append(ii) else: ix_nonout.append(ii) ix_outliers = np.asarray(ix_outliers) # Plot envelope of all non-outlying data lower_nonout = data[ix_nonout, :].min(axis=0) upper_nonout = data[ix_nonout, :].max(axis=0) ax.fill_between(xdata, lower_nonout, upper_nonout, color=plot_opts.get('c_outer', (0.75, 0.75, 0.75))) # Plot central 50% region ax.fill_between(xdata, lower, upper, color=plot_opts.get('c_inner', (0.5, 0.5, 0.5))) # Plot median curve ax.plot(xdata, median_curve, color=plot_opts.get('c_median', 'k'), lw=plot_opts.get('lw_median', 2)) # Plot outliers cmap = plot_opts.get('cmap_outliers') for ii, ix in enumerate(ix_outliers): label = str(labels[ix]) if labels is not None else None ax.plot(xdata, data[ix, :], color=cmap(float(ii) / (len(ix_outliers)-1)), label=label, lw=plot_opts.get('lw_outliers', 1)) if plot_opts.get('draw_nonout', False): for ix in ix_nonout: ax.plot(xdata, data[ix, :], 'k-', lw=0.5) if labels is not None: ax.legend() return fig, depth, ix_depth, ix_outliers
[docs]def rainbowplot(data, xdata=None, depth=None, method='MBD', ax=None, cmap=None): """ Create a rainbow plot for a set of curves. A rainbow plot contains line plots of all curves in the dataset, colored in order of functional depth. The median curve is shown in black. Parameters ---------- data : sequence of ndarrays or 2-D ndarray The vectors of functions to create a functional boxplot from. If a sequence of 1-D arrays, these should all be the same size. The first axis is the function index, the second axis the one along which the function is defined. So ``data[0, :]`` is the first functional curve. xdata : ndarray, optional The independent variable for the data. If not given, it is assumed to be an array of integers 0..N-1 with N the length of the vectors in `data`. depth : ndarray, optional A 1-D array of band depths for `data`, or equivalent order statistic. If not given, it will be calculated through `banddepth`. method : {'MBD', 'BD2'}, optional The method to use to calculate the band depth. Default is 'MBD'. ax : AxesSubplot, optional If given, this subplot is used to plot in instead of a new figure being created. cmap : Matplotlib LinearSegmentedColormap instance, optional The colormap used to color curves with. Default is a rainbow colormap, with red used for the most central and purple for the least central curves. Returns ------- Figure If `ax` is None, the created figure. Otherwise the figure to which `ax` is connected. See Also -------- banddepth, fboxplot References ---------- [1] R.J. Hyndman and H.L. Shang, "Rainbow Plots, Bagplots, and Boxplots for Functional Data", vol. 19, pp. 29-25, 2010. Examples -------- Load the El Nino dataset. Consists of 60 years worth of Pacific Ocean sea surface temperature data. >>> import matplotlib.pyplot as plt >>> import statsmodels.api as sm >>> data = sm.datasets.elnino.load() Create a rainbow plot: >>> fig = plt.figure() >>> ax = fig.add_subplot(111) >>> res = sm.graphics.rainbowplot(data.raw_data[:, 1:], ax=ax) >>> ax.set_xlabel("Month of the year") >>> ax.set_ylabel("Sea surface temperature (C)") >>> ax.set_xticks(np.arange(13, step=3) - 1) >>> ax.set_xticklabels(["", "Mar", "Jun", "Sep", "Dec"]) >>> ax.set_xlim([-0.2, 11.2]) >>> plt.show() .. plot:: plots/graphics_functional_rainbowplot.py """ fig, ax = utils.create_mpl_ax(ax) if cmap is None: from matplotlib.cm import rainbow_r cmap = rainbow_r data = np.asarray(data) if xdata is None: xdata = np.arange(data.shape[1]) # Calculate band depth if required. if depth is None: if method not in ['MBD', 'BD2']: raise ValueError("Unknown value for parameter `method`.") depth = banddepth(data, method=method) else: if depth.size != data.shape[0]: raise ValueError("Provided `depth` array is not of correct size.") ix_depth = np.argsort(depth)[::-1] # Plot all curves, colored by depth num_curves = data.shape[0] for ii in range(num_curves): ax.plot(xdata, data[ix_depth[ii], :], c=cmap(ii / (num_curves - 1.))) # Plot the median curve median_curve = data[ix_depth[0], :] ax.plot(xdata, median_curve, 'k-', lw=2) return fig
[docs]def banddepth(data, method='MBD'): """ Calculate the band depth for a set of functional curves. Band depth is an order statistic for functional data (see `fboxplot`), with a higher band depth indicating larger "centrality". In analog to scalar data, the functional curve with highest band depth is called the median curve, and the band made up from the first N/2 of N curves is the 50% central region. Parameters ---------- data : ndarray The vectors of functions to create a functional boxplot from. The first axis is the function index, the second axis the one along which the function is defined. So ``data[0, :]`` is the first functional curve. method : {'MBD', 'BD2'}, optional Whether to use the original band depth (with J=2) of [1]_ or the modified band depth. See Notes for details. Returns ------- ndarray Depth values for functional curves. Notes ----- Functional band depth as an order statistic for functional data was proposed in [1]_ and applied to functional boxplots and bagplots in [2]_. The method 'BD2' checks for each curve whether it lies completely inside bands constructed from two curves. All permutations of two curves in the set of curves are used, and the band depth is normalized to one. Due to the complete curve having to fall within the band, this method yields a lot of ties. The method 'MBD' is similar to 'BD2', but checks the fraction of the curve falling within the bands. It therefore generates very few ties. The algorithm uses the efficient implementation proposed in [3]_. References ---------- .. [1] S. Lopez-Pintado and J. Romo, "On the Concept of Depth for Functional Data", Journal of the American Statistical Association, vol. 104, pp. 718-734, 2009. .. [2] Y. Sun and M.G. Genton, "Functional Boxplots", Journal of Computational and Graphical Statistics, vol. 20, pp. 1-19, 2011. .. [3] Y. Sun, M. G. Gentonb and D. W. Nychkac, "Exact fast computation of band depth for large functional datasets: How quickly can one million curves be ranked?", Journal for the Rapid Dissemination of Statistics Research, vol. 1, pp. 68-74, 2012. """ n, p = data.shape rv = np.argsort(data, axis=0) rmat = np.argsort(rv, axis=0) + 1 # band depth def _fbd2(): down = np.min(rmat, axis=1) - 1 up = n - np.max(rmat, axis=1) return (up * down + n - 1) / comb(n, 2) # modified band depth def _fmbd(): down = rmat - 1 up = n - rmat return ((np.sum(up * down, axis=1) / p) + n - 1) / comb(n, 2) if method == 'BD2': depth = _fbd2() elif method == 'MBD': depth = _fmbd() else: raise ValueError("Unknown input value for parameter `method`.") return depth